Partial Differential Equations and Diffusion Processes James Nolen 1 Department of Mathematics, Stanford University 1 Email: nolen@math.stanford.edu. Reproduction or distribution of these notes without the author s permission is prohibited.
Contents 1 Introduction to PDE 3 1.1 Notation........................................... 3 1.2 Examples.......................................... 4 1.3 Solutions and Boundary Conditions............................ 5 1.4 Linear vs. Nonlinear.................................... 7 1.5 Some Important Questions................................ 7 2 The Heat Equation, Part I 9 2.1 Physical Derivation and Interpretation.......................... 9 2.2 The Heat Equation and a Random Walker........................ 11 2.3 The Fundamental Solution................................. 14 2.4 Duhamel s principle.................................... 18 2.5 Boundary Value Problems................................. 22 2.6 Uniqueness of solutions: the energy method....................... 25 2.7 Properties of Solutions................................... 27 2.8 Non-constant coefficients.................................. 29 3 Stochastic Processes and Stochastic Calculus 31 3.1 Continuous Time Stochastic Processes and Brownian Motion............. 31 3.1.1 Measure theory................................... 31 3.1.2 Conditional Expectation.............................. 32 3.1.3 Brownian Motion.................................. 34 3.1.4 Properties of Brownian Motion.......................... 35 3.1.5 Stopping times................................... 38 3.2 The Itô Integral and Stochastic Differential Equations................. 38 3.3 Itô s Formula........................................ 44 3.4 Girsanov Transformation.................................. 47 4 Second order linear PDEs and the Feynman-Kac Formula 5 4.1 The Feynman-Kac Formula................................ 5 4.1.1 Poisson s equation................................. 54 4.2 Boundary Value Problems................................. 55 4.3 Transition Densities.................................... 57 5 The Heat Equation, Part II 6 1
CONTENTS Math 227, J. Nolen 5.1 Separation of Variables and Eigenfunction Expansion.................. 6 5.2 Fourier Series........................................ 63 5.3 Solving the heat equation................................. 67 6 The Black-Scholes Equation 72 6.1 Deriving the Black-Scholes equation via risk-neutral pricing.............. 73 6.2 Transformation to heat equation and dimensionless variables............. 76 7 Volatility Estimation and Dupire s Equation 82 7.1 Dupire s equation...................................... 83 7.2 Model Calibration with Dupire s Equation........................ 88 8 Optimal Control and the HJB equation 93 8.1 Deterministic Optimal Control.............................. 93 8.2 The Dynamic Programming Principle.......................... 96 8.3 The Hamilton-Jacobi-Bellman Equation......................... 97 8.4 Infinite Time Horizon................................... 11 8.5 Brief Introduction to Stochastic Optimal Control.................... 12 8.5.1 Dynamic Programming Principle for Stochastic Control............ 14 8.5.2 HJB equation.................................... 14 8.5.3 Examples...................................... 15 2
Chapter 1 Introduction to PDE An ordinary differential equation (ODE) is an equation involving an unknown function y(t), its ordinary derivatives, and the (single) independent variable. For example: y (t) = 3y(t) + t. A partial differential equation (PDE) is an equation involving an unknown function, its partial derivatives, and the (multiple) independent variables. PDE s are ubiquitous in science and engineering; the unknown function might represent such quantities as temperature, electrostatic potential, value of a financial security, concentration of a material, velocity of a fluid, displacement of an elastic material, population density of a biological species, acoustic pressure, etc. These quantities may depend on many variables, and one would like to understand how the unknown quantity depends on these variables. Typically a partial differential equation can be derived from physical laws (like Newton s laws of motion) and/or modeling assumptions (like the no-arbitrage principle) that specify the relationship between the unknown quantity and the variables on which it depends. So, often we are given a model in the form of a PDE which embodies physical laws and modeling assumptions, and we want to find a solution and study its properties. 1.1 Notation For a given domain Ω R d, and a function u : Ω R we will use u x1, u x2, u x2 x 2 u x1 x 1, u x1 x 2... to denote the partial derivatives of u with respect to independent variables x = (x 1,..., x d ) R d. We may also write u u,, 2 u x 1 x 2 x 2, 2 u 2 u 2 x 2,,..., (1.1) 1 x 1 x 2 respectively. When d = 2, we may use (x, y) instead of (x 1, x 2 ), and we may use u x, u y, u yy u xx, u xy,... or u x, u y, 2 u y 2, 2 u x 2, 2 u x y,... (1.2) to denote the partial derivatives. For much of the course, we will consider equations involving a variable representing time, which we denote by t. In this case, we will often distinguish the temporal domain from the domain for the other variables, and we generally will use Ω R d to refer to the domain for the other variables only. For example, if u = u(x, t), the domain for the function u is a subset of R d+1, perhaps Ω R or Ω [, ). In many physical applications, the other variables represent spatial coordinates. For 3
1.2. EXAMPLES Math 227, J. Nolen example, u(x, t) might represent the temperature at position x at time t. In financial applications, however, the variable x might represent non-spatial quantities, like the price of a call option, and in this case we might use different notation. For example, C(s, t) may denote the value of a European call option if the underlying stock price is s at time t. Even though the stock price s does not correspond to physical space, I will typically use the term spatial variables to refer to all of the independent variables except the variable representing time. In fact, one of the most fascinating points of this course is that the Black-Scholes equation for the price of a European call option can be transformed to the heat equation which models the dissipation of heat in a physical body, even though heat transfer and the fluctuation of stock prices are very different phenomena. Normally we will use the notation Du or u to refer to the gradient of u with respect to the spatial variables only. So, Du is the vector Du = (u x1, u x2,..., u xd ). We will use D 2 u to refer to the collection of second partial derivatives of u with respect to x. The term u will always refer to the Laplacian with respect to the spatial coordinates. So, if u = u(x, t) and x R d, u = d u xj x j = j=1 d j=1 2 u x 2 j (1.3) Some people use 2 u for the Laplacian instead of u. In some PDE literature it is common to use multi-indices to denote partial derivatives. A multi-index is a vector α = (α 1, α 2,..., α n ) with integers α i >, and we define D α α u u = x α 1 1 (1.4) xαn n where α = α 1 + + α n is the order of the index. The notation D k u is used to refer to the collection of k th -order partial derivatives of u. 1.2 Examples The order of a PDE is the degree of the highest order derivatives appearing in the equation. For example, if there are two independent variables (x, y), a second-order PDE has the general form while a first-order PDE has the general form F (u xx, u yy, u xy, u x, u y, u, x, y) =. F (u x, u y, u, x, y) =. In multi-index notation these could be written as F (D 2 u, Du, u, x, y) = or F (Du, u, x, y) =. In this course we will discuss only first and second order equations. Here are some examples of second-order equations: The heat equation Black-Scholes equation u t = u xx + u yy (or, u t = u) (2.5) u t = 1 2 σ2 (x, t)x 2 u xx rxu x + ru (2.6) 4
1.3. SOLUTIONS AND BOUNDARY CONDITIONS Math 227, J. Nolen Dupire s equation Laplace s equation Poisson s equation Reaction-diffusion equation The wave equation u t = 1 2 σ2 (x, t)x 2 u xx rxu x (2.7) u xx + u yy = (or, u = ) (2.8) u xx + u yy = f(x, y) (or, u = f) (2.9) u t = u + f(u) (2.1) u tt c 2 u xx = (or, u tt = u) (2.11) Here are a few first order equations: Transport equation Burgers equation A Hamilton-Jacobi equation u t + v(x) Du = (2.12) u t + uu x = (2.13) u t + Du 2 = (2.14) All of these examples involve a single function u. There are also many interesting systems of equations involving multiple unknown quantities simultaneously. For example, the famous Navier- Stokes equations are a second-order system of equations that model the velocity u = {u (i) (x, t)} d i=1 of an incompressible fluid: t u (i) + u u (i) = ν u (i) p, i = 1,..., d u x (i) i = (2.15) 1.3 Solutions and Boundary Conditions i We say that a function u solves a PDE if the relevant partial derivatives exist and if the equation holds at every point in the domain when you plug in u and its partial derivatives. For example, the function u(x, y) = e x cos(y) solves the PDE u xx + u yy = in the domain Ω = R 2. This definition of a solution is often called a classical solution, and we will use this definition unless stated otherwise. However, not every PDE has a solution in this sense, and it is sometimes useful to define a notion of weak solution. The independent variables vary in a domain Ω, which is an open set that may or may not be bounded. A PDE will often be accompanied by boundary conditions, initial conditions, or terminal conditions that prescribe the behavior of the unknown function u at the boundary of 5
1.3. SOLUTIONS AND BOUNDARY CONDITIONS Math 227, J. Nolen the domain under consideration. There are many boundary conditions, and the type of condition used in an application will depend on modeling assumptions. Not all solutions to the PDE will satisfy the boundary conditions. For example, there are many smooth functions that satisfy the Laplace equation (2.8) for all (x, y) in a given smooth domain Ω. However, if we require that a solution u also satisfies the boundary condition u(x, y) = g(x, y) for all points (x, y) in the boundary of the domain, with a prescribed function g(x, y), then the solution will be unique. This is similar to the situation with ODEs: the equation f (x) = f(x) for x has many solutions (f(x) = Ce x ), but if we require that f() = 27, there is a unique solution f(x) = 27e x. The boundary of a domain will be denoted by Ω. One example of a boundary value problem (BVP) for the Laplace equation might be: u xx + u yy =, for all (x, y) Ω (PDE) u(x, y) = g(x, y), for all (x, y) Ω (boundary condition). (3.16) Prescribing the value of the solution u = g on the boundary is called the Dirichlet boundary condition. If g =, we say that the boundary condition is homogeneous. There are many other types of boundary conditions, depending on the equation and on the application. Some boundary conditions involve derivatives of the solution. For example, instead of u = g(x, y) on the boundary, we might impose ν u = g(x, y) for all (x, y) Ω. Here, the vector ν = ν(x, y) is the exterior unit normal vector. This is called the Neumann boundary condition. An example of an initial value problem (IVP) for the heat equation might be: u t = u xx + u yy, for all (x, y, t) Ω (, ) (PDE) u(x, y, t) = g(x, y), for all (x, y, t) Ω (, ) (boundary condition) u(x, y, ) = h(x, y), for all (x, y) Ω (initial condition). (3.17) An initial condition is really a boundary condition on the d + 1 dimensional domain, the spacetime domain. Since t will be interpreted as time, we use the term initial condition to refer to a boundary condition imposed on u at an initial time, often t =. There are also applications for which it is interesting to consider terminal conditions imposed at a future time. For example, terminal value problems (TVP) arise in finance because a financial contract made at the present time may specify a certain payoff at a future time. The Black-Scholes model for the price of a European call option is the terminal value problem u t = 1 2 σ2 x 2 u rxu x + ru, t < T u(x, T ) = max(, x K) (3.18) Here K is the strike price of the option which expires at time T (in the future). Another example of a terminal value problem is the following Hamilton-Jacobi equation: u t + u 2 =, for all x R d, t < T (3.19) u(x, T ) = g(x), for all x R d. (3.2) Solving a BVP means finding a function that satisfies both the PDE and the boundary conditions. In many cases we cannot find an explicit representation for the solution, so solving the problem sometimes means showing that a solution exists or approximating it numerically. 6
1.4. LINEAR VS. NONLINEAR Math 227, J. Nolen 1.4 Linear vs. Nonlinear An equation is said to be linear if it has the form a α (x)d α u = f(x) (4.21) α k (linear combination of u and its derivatives) = (function of the independent variables) A semilinear equation has the form a α (x)d α u + a (D k 1 u,..., Du, u, x) = f(x) (4.22) α =k A quasilinear equation has the form a α (D k 1 u,..., Du, u, x)d α u + a (D k 1 u,..., Du, u, x) = f(x) (4.23) α =k An equation that depends in a nonlinear way on the highest order derivatives is called fully nonlinear. For example, the Laplace equation (2.8), the Poisson equation (2.9), the heat equation (2.5), Dupire s equation (2.7), Black-Scholes equation (.1), the wave equation (2.11), and the transport equation (2.12) are all linear equations. The reaction-diffusion equation (2.1) is semilinear. Burgers equation (2.13) is a quasilinear equation. The Hamilton-Jacobi equation (3.19) is fully nonlinear. If an equation is linear and f in the expression (4.21), the equation is called homogeneous. Otherwise it is called inhomogeneous. For example, the Laplace equation (2.8) is homogeneous, while the Poisson equation (2.9) is the inhomogeneous variety. Generally linear equations are easier than nonlinear equations. One reason for this is that if u and v solve a linear, homogeneous equation, then so does any linear combination of u and v. For example, if u and v both solve the heat equation u t = u xx and v t = v xx then the function w(x, t) = αu(x, t) + βv(x, t) also solves the heat equation. This fact will be used many times in our analysis of linear equations. If the equation were nonlinear, it might not be so easy to find a nice PDE solved by w. 1.5 Some Important Questions Here are some important questions to consider when studying a PDE. Is there a solution? This is not always easy to answer. Sometimes there may not be a classical solution, so one tries to find a weak solution by relaxing some conditions in the problem or weakening the notion of solution. For some problems, we may be able to construct a solution only in a region near part of the domain boundary. For time-dependent problems, sometimes there may be a classical solution only for a short period of time. The ODE analog is what happens to the solution of the problem u = u 2, u() = 1: it has a unique solution u(t) = 1/(1 t) that remains finite only until the time t = 1. If so, is the solution unique? Often a PDE will have infinitely many solutions, but only one of them satisfies the boundary condition under consideration. Nevertheless, verifying uniqueness of the solution may be difficult, especially when the equation is nonlinear. 7
1.5. SOME IMPORTANT QUESTIONS Math 227, J. Nolen Does the solution depend continuously on the data (e.g. boundary conditions, initial conditions, terminal conditions)? That is, if we vary the data a little, does the solution behave in a stable manner or will it change completely? This is a very important property to have if you want to approximate a solution numerically, or if you want to quantify uncertainty in a simulation when there may be small fluctuations in the data. How can we represent the solution? Often we cannot find an explicit formula for the solution, even if the solution is unique. How regular is the solution? Regularity refers to the smoothness of the solution with respect to the independent variables. If the data in the problem have a certain regularity (for instance, is twice differentiable), what can we say about the solution? Will the solution be smoother than the data? Will the solution lose regularity in time? Both can happen: for instance, the heat equation has a regularizing effect: solution is better than the data, while Hamilton-Jacobi equations have solutions that may form singularities even if the data is smooth. Regularity also has practical implications. Roughly speaking, the ability to efficiently and accurately numerically approximate a solution to a PDE is directly related to the solution s regularity: the smoother the solution the easier it is to obtain it numerically. What are the qualitative/quantitative properties of the solution? change as parameters in the equation change? How does the solution How can we approximate the solution numerically? There is no universal numerical method that can be used for every PDE. Our theoretical understanding of the solutions helps in the development of efficient and convergent numerical solution methods. Given a solution (or measurements of a real system modeled by a particular PDE) can one reconstruct parameters in the equation? For instance, can we recover the heat conductivity if we measure the solution of the heat equation?this is an inverse problem. 8
Chapter 2 The Heat Equation, Part I References: Evans, Section 2.3 Strauss, 2.3-2.5, 3.3, 3.5 2.1 Physical Derivation and Interpretation For x R d and t R, the heat equation is u t = u (1.1) In the case of one dimension spatial dimension, d = 1, this is just u t = u xx. The heat equation models diffusion or heat transfer in a system out of equilibrium. The function u(x, t) might represent temperature or the concentration of some substance, a quantity which may vary with time t. Here is a derivation of the equation based on physical reasoning. Let F (u) denote the flux of the quantity represented by u; the flux is a vector quantity representing the flow per unit surface area per unit time. From time t 1 to time t 2, the net change in the amount of u in a region D R d is determined by the net flux through the boundary D. This is a conservation assumption no material or heat energy is created or destroyed. This means that for any t 1 < t 2, D t2 u(x, t 2 ) dx u(x, t 1 ) dx = F (u) ν dx dt. D t 1 D The flux may be modeled as a linear function of u: F (u) = a u, where a > is a constant. If u represents temperature, this assumption is known as Fourier s law of heat conduction; if u represents the concentration of a diffusing material, this is known as Fick s law. Therefore, the 9
2.1. PHYSICAL DERIVATION AND INTERPRETATION Math 227, J. Nolen function u(x, t) should satisfy D t2 u(x, t 2 ) dx u(x, t 1 ) dx = D = = = t 1 t2 t 1 t2 t 1 t2 t 1 D D D D F (u) ν dx dt F (u) dx dt (a u) dx dt (the conservation assumption) (using the Divergence theorem) (assuming F (u) = a u) a u dt dx (1.2) Taking D = B r (x), a ball of radius r centered at a point x, dividing by D, and letting r, we conclude that u must satisfy u(x, t 2 ) u(x, t 1 ) = t2 t 1 a u(x, t) dt (1.3) for all x. This is just the integral (in time) form of the heat equation (1.1). Here we have used the fact that if w(x, t) is a continuous function, then 1 lim w(y, t) dy = w(x, t) (1.4) r B r (x) B r(x) where B r (x) denotes the ball of radius r centered at the point x, and B r (x) denotes the volume of the ball. If there is an external volumetric source (heat source, injection of material, etc.) or sink (cold bath, depletion of material, etc.) represented by a function f(x, t), then we have D t2 t2 u(x, t 2 ) dx u(x, t 1 ) dx = F (u) ν dx dt + f(x, t) dx dt. D t 1 D t 1 D and the equation for u becomes inhomogeneous: u t = a u + f(x). The case f > represents an inflow of material or a heat source. The case f < models outflow of material or a heat sink. In general f might not have a constant sign, representing the presence of both sources and sinks. In some models, the source or sink might depend on u itself: f = f(u). For example, a simple model of an exothermic reaction might be f = cu, where the parameter c > models a reaction rate. In physical applications, the parameter a > is sometimes called the thermal conductivity or the diffusivity. Notice that large values of a model rapid heat transfer or rapid diffusion; small values of a model slow heat transfer or slow diffusion. In some applications, the constant a is replaced by a matrix a ij (x) modeling a situation where the conductivity is variable, as in a composite material, for example. In this case, the derivation above produces the equation u t = (a(x) u) + f (1.5) 1
2.2. THE HEAT EQUATION AND A RANDOM WALKER Math 227, J. Nolen The simple heat equation corresponds to a ij (x) Id (the identity matrix) and f. Assuming that a(x) is differentiable, we could also write this as u t = i,j a ij (x)u xi x j + b(x) u + f (1.6) where b(x) = (b j (x)), b j (x) = i x i a ij (x). This equation is said to be in non-divergence form, while the equation (1.5) is said to be in divergence form. As we will see later, the Black-Scholes equation has a similar form: u t = 1 2 σ2 (x, t)x 2 u xx rxu x + ru (1.7) Here the coefficient in front of the u xx term might depend on both x and t, in general. We may consider the heat equation for x R d, or in a bounded domain Ω R d with appropriate boundary conditions. For example, we will consider the initial value problem with Dirichlet boundary condition u t = u(x) + f(x, t), x Ω, t > u(x, t) = h(x, t), x Ω, t > u(x, ) = g(x), x Ω, t = (1.8) where Ω R d is some smooth, bounded domain. The Dirichlet boundary condition u(x, t) = h(x, t) on Ω may be interpreted as fixing the temperature at the boundary. Alternatively, the Neumman boundary condition ν u(x, t) = g(x, t) corresponds to prescribing the heat flux at the boundary (perhaps via an insulating layer, which means that g = ). 2.2 The Heat Equation and a Random Walker The preceding derivation of the heat equation was based on physical reasoning. Here is a rather different derivation and interpretation of the heat equation which illuminates its connection with probability and stochastic processes. Consider a simple random walk X x (n) on the integers Z. We suppose that at each time step, the process moves independently either to the left, or to the right with probability 1/2, and that X x () = x initially. That is X x (n) = x + n s j (2.9) where the steps s j are independent, identically distributed random variables with P (s j = +1) = 1/2 and P (s j = 1) = 1/2 for each j. j=1 The exit time Let us assume that the starting point x Z lies between two integers a, b Z: a x b. Consider the random time s(x) that the walker spends before it hits either a or b if it started at x, and let 11
2.2. THE HEAT EQUATION AND A RANDOM WALKER Math 227, J. Nolen τ(x) be its expected value: τ(x) = E(s(x)). As the walker moves initially either to the right or to the left,with equal probabilities, and spends a unit time to do so, we have the simple relation τ(x) = 1 2 τ(x 1) + 1 τ(x + 1) + 1, (2.1) 2 that may be re-written in the form of a discrete Poisson equation: τ(x + 1) + τ(x 1) 2τ(x) = 1, (2.11) 2 which is supplemented by the boundary conditions τ(a) = τ(b) =. The expected prize value Now suppose that f(x) is a prescribed smooth function and that after n steps, we evaluate the expectation u(x, n) = E [f(x x (n))]. You may think of u(x, n) as the expected payoff for a random walker starting at point x and walking for time n. The prize for landing at point X x (n) = y at time n is f(y), and u(x, n) is the expected prize value. How does u depend on the starting point x, time n, and prize distribution f? At the initial time n =, the walker has not moved from its starting point, so we must have u(x, ) = E [f(x x ())] = f(x). Now consider n >. Since X x (n) is a Markov process, we see that E [f(x x (n))] = E [E(f(X x (n)) X x (n 1))] for each n >. The term E(f(X x (n)) X x (n 1)) is the conditional expectation of the payoff at time n, given the position at time n 1. We also know that starting from the point X x (n 1), the walker then moves to either X x (n 1) 1 (left) or to X x (n 1) + 1 (right), each with probability 1/2. Therefore, we can evaluate this conditional expectation explicitly: E[f(X x (n)) X x (n 1)] = 1 2 f(x x(n 1) 1) + 1 2 f(x x(n 1) + 1). Taking the expected value on both sides, we conclude that E [f(x x (n))] = E [E(f(X x (n)) X x (n 1))] = 1 2 E[f(X x(n 1) 1)] + 1 2 E[f(X x(n 1) + 1)] Now subtract E [f(x x (n 1))] from both sides and we find that u(x, n) u(x, n 1) = E [f(x x (n))] E [f(x x (n 1))] = 1 2 E[f(X x(n 1) 1)] + 1 2 E[f(X x(n 1) + 1)] E [f(x x (n 1))] Observe that X x (n 1) ± 1 = X x±1 (n 1) almost surely. This observation allows us to express the change of u with respect to n in terms of changes in u with respect to x: u(x, n) u(x, n 1) = 1 2 E[f(X x(n 1) 1)] + 1 2 E[f(X x(n 1) + 1)] E [f(x x (n 1))] = 1 2 E[f(X x 1(n 1))] + 1 2 E[f(X x+1(n 1))] E [f(x x (n 1))] = 1 (u(x 1, n 1) 2u(x, n 1) + u(x + 1, n 1)) (2.12) 2 12
2.2. THE HEAT EQUATION AND A RANDOM WALKER Math 227, J. Nolen Observe! The relationship (2.12) looks very much like a discrete version of the heat equation! Let us explore this further. If we had let the spatial jumps be of size h > instead of size 1 and let the time steps be of size δ > instead of size 1, the same derivation would lead us to the equation u(x, t) u(x, t δ) = 1 (u(x h, t δ) 2u(x, t δ) + u(x + h, t δ)) 2 Here we use t δz to denote a point of the form t = nδ for some integer n. Now suppose we make the clever choice δ = h 2 /2. Then after dividing both sides by δ we have u(x, t) u(x, t δ) δ Taylor s theorem tells us that for differentiable v, = (u(x h, t δ) 2u(x, t δ) + u(x + h, t δ)) h 2 (2.13) Also, v(x, t) v(x, t δ) δ = v t (x, t) + O(δ). v(x + h) = v(x) + hv x (x) + h2 2 v xx(x) + O(h 3 ) v(x h) = v(x) hv x (x) + h2 2 v xx(x) + O(h 3 ) so that v(x + h) 2v(x) + v(x h) h 2 = v xx (x) + O(h 2 ). From this we see that (2.13) is a discrete version of the equation u t = u xx. This suggests that if we let the step size h (with δ = h 2 /2), the function u (which depends on h) converges to some function v(x, t) which satisfies v t = v xx. How does the random walk behave under this scaling limit? t = kδ, then the scaled random walk may be written as j=1 If δ = 1/N, h = 2/ N, and 2 Nt X x (t) = x + s j. (2.14) N For continuous time t >, define the piecewise linear function X x (t) by interpolating the points {(n, X x (n))} n like this: X x (t) = (n + 1 t)x x (n) + (t n)x x (n + 1), if t [n, n + 1). (2.15) Since E[s j ] = and E[s 2 j ] = 1, the functional central limit theorem implies that the process X x (t) converges weakly, as h, to 2W x (t) where W x (t) is a Brownian motion with W x () = x (for example, see Karatsas and Shreve, section 2.4). Therefore, for any fixed t, Xx (t) converges weakly to a Gaussian random variable with mean µ = x and variance σ 2 = 2t. Hence, lim E[f(X x(t))] = h 13 R 1 4πt e x y 2 4t f(y) dy (2.16)
2.3. THE FUNDAMENTAL SOLUTION Math 227, J. Nolen In summary, we have observed that in the limit of small step sizes, the value function u(x, t) = E[f(X x (t))] representing the payoff at time t for a walk started at x satisfies the heat equation with initial data u(x, ) = f(x). Alternatively, the formula u(x, t) = E[f(X x (t))] identifies the solution to the initial value problem for the heat equation u t = u xx with a functional of Brownian motion. Later we will use Itô s formula to study this connection in much more generality. For now, notice how the following ideas appeared in this formal derivation: The Markov property of the random walk The fact that the walker steps to the left or right with equal probability (related to the Martingale property) The use of the space-time scaling ratio δ/h 2 = constant Exercises: 1. Perform similar analysis for dimension d = 2. That is, show that the expected payoff E[f(X x,y (n))] for the a random walk on the lattice Z 2 satisfies a discrete version of the heat equation u t = u xx + u yy. 2. Consider a random walker on integers with a bias: it jumps to the left with probability p and to the right with a probability (1 p). What are the discrete equations for the exit time and the prize distribution? Find their continuous limits as well. 3. Consider a random walker on Z 2 with asymmetry: assume that probability to go up or down is 1/4 p while the probability to go left or right is 1/4 + p. What is the corresponding discrete equation for the exit time from a square on the integer lattice, and what is the continuous limit? 2.3 The Fundamental Solution Observe that the heat equation is a linear equation. Therefore, if u and v are both solutions to the heat equation, then so is any linear combination of u and v. This fact will be used frequently in our analysis. Here we define a very special solution which allows us to construct solutions to initial value problems. The fundamental solution for the heat equation is the function Φ(x, t) = 1 x 2 e 4t (4πt) d/2, (3.17) defined by t >. We have already seen this function in our derivation of the relation (2.16). Now we take this function as the starting point and show how it can be used to solve the heat equation. This function is also called the heat kernel, and it has the following properties: (P) For t >, Φ(x, t) > is an infinitely differentiable function of x and t. (P1) Φ t = Φ for all x R d and t >. (P2) R d Φ(x, t) dx = 1 for all t >. Also, for each t >, Φ(x, t) is the probability density for a multivariate Gaussian random variable x R d with mean µ = and covariance matrix Σ ij = 2tδ ij (in one dimension, σ 2 = 2t). 14
2.3. THE FUNDAMENTAL SOLUTION Math 227, J. Nolen (P3) For any function g(x) that is continuous and satsifies g(x) C 1 e C2 x, for some C 1, C 2 >, lim Φ(x, t)g(x) dx = g(). t R d In particular, this holds for any continuous and bounded function. Property P1 is easy if slightly tediously to verify directly by taking deriviatives of Φ(x, t). Property P2 says that the integral of Φ is invariant in t (remember, no heat created or destroyed). This is easy to verify by using a change of variables and the following basic fact: ( ( ) 1/2 so that e x2 dx = = ( 1 Φ(x, t)dx = R d (4πt) d/2 = 1 ( π d/2 ) 2 1/2 e dx) x2 = 2π R d e re r2 dθ dr) 1/2 = ( e x2 y 2 dx dy ( 2π re r2 dr) 1/2 = π, (3.18) x 2 1 4t dx = π d/2 e y2 dy = 1 R d π d/2 e y2 1 y2 2 y2 ddy R d d e dz) z2 = 1. R Because the integral of Φ(x, t) > is 1 for all t >, the function Φ(x, t) defines a probability density for each t > fixed. In fact, this is just the density for a multivariate Gaussian random variable with mean zero and covariance matrix Σ ij = 2tδ ij (in one dimension, σ 2 = 2t). So as t, the variance grows linearly, and the standard deviation is proportional to t. Property P3 is a very interesting property which says that as t the function Φ(x, t) concentrates at the origin. If Φ represents the density of a diffusing material at point x at time t, then P3 says that all of the mass concentrates at x = as t. Mathematically this means that that Φ converges to a Dirac delta function (δ ) in the sense of distributions as t. Since Φ(x, t) >, you may think of the integral Φ(x, t)g(x) dx (3.19) R d as a weighted average of the function g(x). In fact, this integral is an expectation with respect to the probability measure defined by Φ. As t, all of the weight concentrates near the origin where g = g(). In order to verify P3 we write: R d Φ(x, t)g(x)dx = 1 (4πt) d/2 R d e x 2 1 4t g(x)dx = π d/2 R d y e 2 g(y 4t)dy 1 y π d/2 e 2 g()dy = g(), R d as t. In the last step we used the Lebesgue Dominated convergence theorem, since for all t (, 1) we have a bound for the integrand by an integrable function independent of t (, 1): e y2 g(y 4t) C 1 e y2 e C 2 y 4t C 1 e y2 +2C 2 y, which is integrable. Using these properties one may show the following: 15
2.3. THE FUNDAMENTAL SOLUTION Math 227, J. Nolen Theorem 2.3.1 For any function g(x) that is continuous and satisfies g(x) C 1 e C2 x for some C 1, C 2 >, the function u(x, t) = Φ(x y, t)g(y) dy (3.2) R d satisfies (i) u C (R d (, )) (u is smooth in x and t for all positive times) (ii) u t = u, for all x R d and t > (iii) lim u(x, t) = g(x ) (3.21) (x,t) (x, + ) So, the function u(x, t) defined by (3.2) solves the initial value problem in R d with initial data g(x). The values at t = are defined by continuity, since the formula (3.2) is ill-defined for t =. Nevertheless, property (iii) says that the limit as t + is well defined and equal to g. Here is a very interesting point: even if g(x) is merely continuous (not necessarily differentiable), we have a solution to the heat equation which is actually infinitely differentiable for all positive times! This is sometimes referred to as the smoothing property of the heat equation. Obviously, not all PDE s have this property. For instance, the simple transport equation u t + u x =, u(x, ) = g(x) has the solution u(x, t) = g(x t) which is not at all smoother than the the initial data. The qualitative difference between the smoothing properties of the heat equation and the transport equation lies in the fact that the heat equation has a genuinely stochastic representation that produces the regularizing effect. Consider the convolution formula: u(x, t) = Φ(x y, t)g(y) dy. R d Since Φ(y, t) is the density for a probability measure, then so is Φ(x y, t), although the mean is shifted to the point x. Therefore, this convolution formula is really an expectation u(x, t) = E [g(x x (t))], where {X x (t)} t denotes a family of normal random variables with mean x and variance 2t. This is precisely the conclusion (2.16) of our earlier derivation of the heat equation using the discrete random walk. Note that if the growth condition g(x) C 1 e C2 x were not satisfied, the integral Φ(x, t)g(x) dx might not even be finite. For example, if g(x) = e x2 the integral is not finite if t is large enough. Proof of Theorem 2.3.1: First, property (iii) is a simple consequence of P3. Indeed, let g x (y) = g(x y), then P3 implies that lim Φ(x y, t)g(y)dy = lim Φ(y, t)g(x y)dy = lim Φ(y, t)g x (y)dy = g x () = g(x ) = g(x). t t t 16
2.3. THE FUNDAMENTAL SOLUTION Math 227, J. Nolen Properties (i) and (ii) follow from the fact that we may take derivatives of u by interchanging integration and differentiation. In general, one cannot do this. However, for t > t >. the function Φ(x, t) is smooth with uniformly bounded and integrable derivatives of all orders (their seize is bounded by constants depending on t ). Therefore, one can compute derivatives, as follows, invoking the dominated convergence theorem. The partial derivative u t is defined by the limit We know that as h, lim h so we d like to say that lim h u t = lim h Φ(x y, t + h) Φ(x y, t) R d h g(y) dy. Φ(x y, t + h) Φ(x y, t) g(y) = Φ t (x y, t)g(y), (3.22) h Φ(x y, t + h) Φ(x y, t) R d h g(y) dy = Φ t (x y, t)g(y) dy R d (3.23) also holds. If h is sufficiently small (say h < ɛ), then t ± h >, and Taylor s theorem implies Φ(x y, t + h) Φ(x y, t) = Φ t (x y, t) + R(x, y, t, h), (3.24) h where the remainder R satisfies the bound Therefore, we see that for each x Φ(x y, t + h) Φ(x y, t) g(y) h (max z R(x, y, t, h) h max s ɛ Φ tt(x y, t + s). (3.25) ( g(z) ) Φ t (x y, t) + ɛ max s ɛ Φ tt(x y, t + s) ) (3.26) By computing Φ t and Φ tt directly, we see that the right hand side of (3.26) is integrable in y. Therefore, the dominated convergence theorem implies that u t = lim Φ(x y, t + h) Φ(x y, t) g(y) dy = Φ t (x y, t)g(y) dy. h R d h R d That is, using the dominated convergence theorem, we may justify bringing the limit inside the integral in (3.23). Using a similar argument with the dominated convergence theorem, one can show that u = Φ(x y, t)g(y) dy, (3.27) R d also holds, so that u t u = (Φ t (x y, t) Φ(x y, t)) g(y) dy = (3.28) R d The last equality holds since Φ is itself a solution to Φ t Φ =. In the same way, using the dominated convergence theorem, one may also take higher derivatives of u(x, t), since Φ is infinitely differentiable, and each derivative is integrable (for t > ). This shows that u(x, t) C (R d (, )), even if the initial data g(x) is not smooth! In the case d = 1, Strauss works this out in section 3.5 (see Theorem 1, p. 79). 17
2.4. DUHAMEL S PRINCIPLE Math 227, J. Nolen 2.4 Duhamel s principle So far we have derived a representation formula for a solution to the homogeneous heat equation in the whole space x R d with given initial data. With the fundamental solution we may also solve the inhomogeneous heat equation using a principle called Duhamel s principle. Roughly speaking, the principle says that we may solve the inhomogeneous equation by regarding the source at time s as an initial condition at time s, an instantaneous injection of heat. The solution u is obtained by adding up (integrating) all of the infinitesimal contributions of this heating. A steady problem converted to time-dependence A simply example of how a steady problem can be converted to a time-dependent one is an elliptic problem of the form (a(x) u) = f(x), for x Ω, (4.29) u(x) = for x Ω. (4.3) This problem is posed in a smooth domain Ω R d, and a(x) is the (possibly varying in space) diffusion coefficient. If a(x) = 1 then (4.29 is the standard Poisson equation u = f. The Dirichlet boundary condition (4.3) mean that the boundary is cold (if we think of u as temperature). Here is how solution of (4.29)-(4.3) may be written in terms of a time-dependent problem without a source. Let φ(t, x) be the solution of the initial-boundary-value problem φ t = (a(x) φ), for x Ω, (4.31) φ(x, t) = for x Ω and all t >, (4.32) φ(x, ) = f(x) for x Ω. (4.33) The function φ(t, x) goes to zero as t +, uniformly in x Ω there are various ways to see that but we will take this for granted at the moment. Consider ū(x) = φ(x, t)dt, (4.34) this function satisfies the boudnary condition ū(x) = for x Ω, and, in addition, if we integrate (4.31) in time from t = to t = + we get f(x) = (a(x) ū), (4.35) which is nothing but (4.31). Hence, ū solves (4.31), so that solution of (4.31) can be represented as in (4.34) in terms of solutions of the initial-boundary-value problem which may be sometimes more convenient to solve than the elliptic problem directly. Furthermore, as in reality φ(x, t) goes to zero exponentially fast in time, a good approximation to ū(x) may be obtained by integration not from to + but rather over a short initial time period [, T ]. 18
2.4. DUHAMEL S PRINCIPLE Math 227, J. Nolen The time-dependent Duhamel s principle Suppose we wish to solve u t = u + f(x, t), x R d, t > (4.36) u(x, ) = g(x), x R d. First, for s, we define the family of functions w(x, t; s) solving w t = w, x R d, t > s w(x, s; s) = f(x, s), x R d, t = s. Notice that for each s, w(, ; s) solves an initial value problem with initial data prescribed at time t = s, instead of t =. Then set w(x, t) = t w(x, t; s) ds. (4.37) So, w(x, t; s) represents the future influence (at time t) of heating at time s (, t), and w(x, t) may be interpreted as the accumulation of all the effects from heating in the past. Duhamel s principle says that the solution u(x, t) of the initial value problem (4.36) is given by u(x, t) = u h (x, t) + w(x, t) = u h (x, t) + t w(x, t; s) ds, (4.38) where u h (x, t) solves the homogeneous problem: u h t = u h, x R d, t > u h (x, ) = g(x), x R d. (4.39) In fact (we will prove below), the function w(x, t) is the solution to the inhomogeneous problem with zero initial data: w t = w + f(x, t), x R d, t > (4.4) w(x, ) =, x R d Since the PDE is linear, the combination of u h and w solves (4.36). Now, by Theorem 2.3.1 we may represent both of the functions w and u h in terms of the fundamental solution. Specifically, w(x, t; s) = Φ(x y, t s)f(y, s) dy R d u h (x, t) = Φ(x y, t)g(y) dy (4.41) R d Combining this with the Duhamel formula (4.38), we see that t u(x, t) = Φ(x y, t)g(y) dy + Φ(x y, t s)f(y, s) dy ds (4.42) R d R d 19
2.4. DUHAMEL S PRINCIPLE Math 227, J. Nolen Theorem 2.4.1 (see Evans Theorem 2, p. 5) Suppose f C 2 1 (Rd [, )), then the function defined by (4.42) satisfies (i) u t = u + f(x, t) for all t >, x R d. (ii) u C 2 1 (Rd [, )) (iii) lim u(x, t) = g(x ). (4.43) (x,t) (x, + ) Proof: By the above analysis and Theorem 2.3.1, the only thing left to prove is that the function w(x, t) = solves the inhomogeneous problem (1.3). We compute derivatives: w(x, t + h) w(x, t) h t = 1 h = 1 h + R d Φ(x y, t s)f(y, s) dy ds (4.44) t+h t+h t t w(x, t + h; s) ds 1 h w(x, t + h; s) ds t w(x, t + h; s) w(x, t; s) h Using the properties of f and Φ and integrating by parts, one can show that and that 1 t+h lim h h t ds w(x, t + h; s) ds w(x, t + h; s) ds = w(x, s; s) = f(x, s) (4.45) t lim h w(x, t + h; s) w(x, t; s) h ds = = t t w t (x, t; s) ds w(x, t; s) ds = w(x, t) (4.46) Therefore, w t = w + f(x, t). The initial condition is satisfied since t t lim t Φ(x y, t s)f(y, s) dy ds lim max f(y) ds =. (4.47) R d t y So, w(x, ) =. See Evans Theorem 2, p. 5 for more details. 2
2.4. DUHAMEL S PRINCIPLE Math 227, J. Nolen Relation to ODE s You may have encountered Duhamel s principle already in the context of inhomogeneous ODEs. Let us point out the formal connection between the results above and what you may have seen already. Consider the following homogeneous ODE: η (t) = Aη(t) (4.48) η() = η. (4.49) Here η : [, ) R, and A R is some positive constant. The solution is the exponential Now consider the inhomogeneous ODE: η(t) = e ta η ζ (t) = Aζ(t) + F (t) (4.5) ζ() = ζ The solution is: Exercise: Verify this. ζ(t) = e ta ζ + t e (t s)a F (s) ds. (4.51) You may think of S(t) = e ta as a solution operator for the homogeneous equation (4.48). It maps the initial point η to the value of the solution of (4.48) at time t: S(t)η e ta η. With this definition, the solution to the inhomogeneous equation (4.51) may be written as ζ(t) = S(t)ζ + t S(t s)f (s) ds. (4.52) Formally, the PDE (4.36) has the same structure as the ODE system (4.5). Letting ζ(t) denote the function u(, t), we may write a formal equation ζ (t) = Aζ + F (t) (4.53) where A is now an operator acting on the function ζ(t) = u(, t) according to A : u(, t) u(, t), and F (t) is the function f(, t). This idea of defining an ODE for a function that takes values in a space of functions can be made mathematically rigorous using semigroup theory (for example, see Evans section 7.4, or the book Functional Analysis by K. Yosida). Suppose that we know the solution operator S(t) for the homogeneous equation η = Aη, corresponding to the homogeneous heat equation u t = u. Then the representation (4.52) suggests that the solution to the inhomogeneous equation should be u(x, t) = S(t)g(x) + t S(t s)f(x, s) ds (4.54) We have already computed the solution operator S(t) it is given by convolution with the heat kernel: S(t)g(x) = Φ(x y, t)g(y) dy (4.55) R d 21
2.5. BOUNDARY VALUE PROBLEMS Math 227, J. Nolen and S(t s)f(x, s) = Φ(x y, t s)f(y, s) dy. R d (4.56) Combining this with (4.54) gives us the solution formula (4.42). 2.5 Boundary Value Problems On the Half-Line We now demonstrate a technique for solving boundary value problems for the heat equation on the half-line: u t = u, x >, t > (5.57) u(x, ) = g(x), x > u(, t) =, t >. The boundary condition imposed at x = is the homogeneous Dirichlet condition. The convolution formula (3.2) gives a solution on the entire line x R, but this function will not necessarily satisfy the boundary condition at x =. In fact, g is not yet defined for x <, so for this problem the convolution formula does not make sense immediately. So, we need to modify our approach to solving the problem. The idea we demonstrate here is to construct a solution on the whole line in such a way that the condition u(, t) = is satisfied for all t. Then the restriction of this function to the right half-line will be a solution to our problem (5.57). To construct a solution on the whole line, we need to define the initial data for x <. The key observation is that if the initial data on the whole line has odd symmetry, then the heat equation preserves this symmetry. Moreover, any continuous function f(x) that has odd symmetry (i.e. f( x) = f(x)) must satisfy f() =. Therefore, if u(x, t) has odd symmetry for all t, then u(, t) = will be satisfied for all t. We begin by extending the function g(x) on (, ) by odd-reflection: g ex (x) = g(x), x, g ex (x) = g( x), x < (5.58) This function has odd symmetry: g ex ( x) = g ex (x). Then we solve the extended problem ū t = ū, x R, t > ū(x, ) = g ex (x), x R Using the convolution formula, our solution is ū(x, t) = Φ(x y, t)g ex (y) dy. R Using a change of variables and the fact that Φ has even symmetry, it is easy to see that ū has odd-symmetry: ū( x, t) = ū(x, t) for all x R. Therefore, ū(, t) = for all t >, and the 22
2.5. BOUNDARY VALUE PROBLEMS Math 227, J. Nolen restriction of ū(x, t) to the half-line satisfies (5.57). So, our solution is (for x ): u(x, t) = ū(x, t) = Φ(x y, t)g ex (y) dy = R 1 (4πt) 1/2 e x y 2 4t g ex (y) dy = 1 (4πt) 1/2 R (e x y 2 4t e x+y 2 4t Inhomogeneous boundary conditions, shifting the data Suppose we modify the above problem to become ) g(y) dy (5.59) u t = u, x >, t > (5.6) u(x, ) = g(x), x > u(, t) = h(t), t > Now the boundary condition at the origin is u(, t) = h(t) which may be non-zero in general. Therefore, the reflection technique won t work without modification, since odd reflection guaranteed that u = at the boundary. One way to solve boundary value problems with inhomogeneous boundary conditions is to shift the data. That is, we subtract something from u that satisfies the boundary condition (but maybe not the PDE). In the present case, suppose we have a function ĥ(x, t) : [, ) [, ) R such that ĥ(, t) = h(t). This function ĥ extends h off the axis x =. Then let v(x, t) = u(x, t) ĥ(x, t). This function v satisfies the homogeneous boundary condition: v(, t) = u(, t) ĥ() = h(t) h(t) =. However, v solves a different PDE. Since u = v + ĥ, we compute t (v + ĥ) = (v + ĥ) (5.61) so that v satisfies Putting this all together, we see that u = v + ĥ where v solves v t = v + ĥ ĥt (5.62) v t = v + f(x, t), x >, t > (5.63) v(x, ) = g(x) ĥ(x, ), x > v(, t) =, t > and f(x, t) = ĥ ĥt. The price to pay for shifting the data is that now we may have an inhomogeneous equation and different initial conditions. The key fact that makes this solution technique possible is the fact that the equation is linear; thus we can easily derive and solve an equation for the shifted function v. 23
2.5. BOUNDARY VALUE PROBLEMS Math 227, J. Nolen Another example Here we illustrate this shifting technique and the reflection technique together. Let us solve the inhomogeneous equation with inhomogeneous Dirichlet boundary condition: u t = u xx + k(x, t), t >, x > u(x, ) = g(x), x > u(, t) = 1, t > We also have added a source term k(x, t) just for illustration. Here we suppose that g(x) is smooth, bounded, and g() = 1. Let s find an integral representation formula for u(x, t) involving g, k, and the fundamental solution Φ(x, t). The boundary condition is inhomogeneous, so we first shift the function u to transform the boundary condition into the homogeneous condition. One way to do this would be setting v(x, t) = u(x, t) 1. Then v(, t) = u(, t) 1 =, so v satisfies the homogenous boundary condition. There are other ways to do this, as well. The function v satisfies the modified problem: v t = v xx + k(x, t), t >, x > v(x, ) = g(x) 1 := g(x), x > v(, t) =, t > We now solve for v and set u = v + 1. To solve for v, we extend the problem onto the entire line and solve using the fundamental solution and Duhamel s principle. To obtain the boundary condition, we extend g and k by odd reflection: g ex (x) = g(x) = g(x) 1, x, g ex (x) = g( x) = 1 g( x), x < k ex (x) = k(x), x, k ex (x) = k( x), x < Therefore, using the Duhamel formula (4.42), we construct a solution so that v(x, t) = R Φ(x y, t) g ex (y) dy + t u(x, t) = 1 + v(x, t) = 1 + Φ(x y, t) g ex (y) dy + R (e x y 2 4t 1 = 1 + (4πt) 1/2 + t 1 (4πs) 1/2 ( e x y 2 4(t s) R t Φ(x y, t s)k ex (y, s) dy ds (5.64) R e x+y 2 4t Φ(x y, t s)k ex (y, s) dy ds (5.65) ) (g(y) 1) dy ) e x+y 2 4(t s) k(y, s) dy ds (5.66) 24
2.6. UNIQUENESS OF SOLUTIONS: THE ENERGY METHOD Math 227, J. Nolen 2.6 Uniqueness of solutions: the energy method Using the fundamental solution we have constructed one solution to the problem u t = u + f(x, t), x R d, t > (6.67) u(x, ) = g(x), x R d where f C 2 1 (Rd [, )) and g(x) C 1 e C 2 x. Is this the only solution? If there were another solution v, then their difference w = u v would satisfy w t = w, x R d, t > (6.68) w(x, ) =, x R d since the equation is linear. We d like to say that w for all t > since the initial data is zero. This would imply that u = v so that the solution is unique. However, it turns out (surprise!) that there are non-trivial solutions to this initial value problem (6.68). So the solution to (6.67) is not unique. Nevertheless, the non-trivial solutions to (6.68) must grow very rapidly as x, and if we restrict our attention to solutions satisfying a certain growth condition, then the only solution of (6.68) is the trivial solution w. Therefore, under a certain growth restriction, the solution to (6.67) must be unique: Theorem 2.6.1 (See Evans Theorem 7, p. 58) There exists at most one classical solution to the initial value problem (6.67) satisfying the growth estimate for constants A, a >. u(x, t) Ae a x 2, x R d, t [, T ] (6.69) From now on, we will always assume that our solutions to the heat equation in the whole space satisfy this growth condition. Notice that the condition g(x) C 1 e C 2 x is within the limits of the this growth condition. For boundary value problems in a bounded domain, this technical issue does not arise, and solutions may be unique. For example, consider the initial value problem with Dirichlet boundary conditions: u t = u + f(x, t), x Ω, t > (6.7) u(x, t) = h(x, t), x Ω, t > u(x, ) = g(x), x Ω, t = Theorem 2.6.2 There is at most one solution to the initial value problem (6.7). Proof: If there were two classical solutions to this problem, then their difference w = u v would satisfy (since the equation is linear!): w t = w, x Ω, t > w(x, t) =, x Ω, t > w(x, ) =, x Ω, t = 25